\(\int (c+d x)^3 \cosh ^3(a+b x) \, dx\) [17]

Optimal result
Mathematica [A] (verified)
Rubi [C] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [B] (verification not implemented)
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 16, antiderivative size = 175 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=-\frac {40 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b} \] Output:

-40/9*d^3*cosh(b*x+a)/b^4-2*d*(d*x+c)^2*cosh(b*x+a)/b^2-2/27*d^3*cosh(b*x+ 
a)^3/b^4-1/3*d*(d*x+c)^2*cosh(b*x+a)^3/b^2+40/9*d^2*(d*x+c)*sinh(b*x+a)/b^ 
3+2/3*(d*x+c)^3*sinh(b*x+a)/b+2/9*d^2*(d*x+c)*cosh(b*x+a)^2*sinh(b*x+a)/b^ 
3+1/3*(d*x+c)^3*cosh(b*x+a)^2*sinh(b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {-486 d \left (2 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)-2 d \left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))+12 b (c+d x) \left (82 d^2+15 b^2 (c+d x)^2+\left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{216 b^4} \] Input:

Integrate[(c + d*x)^3*Cosh[a + b*x]^3,x]
 

Output:

(-486*d*(2*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] - 2*d*(2*d^2 + 9*b^2*(c + 
d*x)^2)*Cosh[3*(a + b*x)] + 12*b*(c + d*x)*(82*d^2 + 15*b^2*(c + d*x)^2 + 
(2*d^2 + 3*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(216*b^4)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.28, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.312, Rules used = {3042, 3792, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 3791, 3042, 3777, 26, 3042, 26, 3118}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^3 \cosh ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {2 d^2 \int (c+d x) \cosh ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \cosh (a+b x)dx-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}-\frac {3 i d \int -i (c+d x)^2 \sinh (a+b x)dx}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \sinh (a+b x)dx}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}-\frac {3 d \int -i (c+d x)^2 \sin (i a+i b x)dx}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \int (c+d x)^2 \sin (i a+i b x)dx}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \int (c+d x) \cosh (a+b x)dx}{b}\right )}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {i d \int -i \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int -i \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}+\frac {i d \int \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {2 d^2 \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \cosh (a+b x)dx-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {i d \int -i \sinh (a+b x)dx}{b}\right )-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int \sinh (a+b x)dx}{b}\right )-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \int -i \sin (i a+i b x)dx}{b}\right )-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sinh (a+b x)}{b}+\frac {i d \int \sin (i a+i b x)dx}{b}\right )-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )-\frac {d \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sinh (a+b x)}{b}+\frac {3 i d \left (\frac {i (c+d x)^2 \cosh (a+b x)}{b}-\frac {2 i d \left (\frac {(c+d x) \sinh (a+b x)}{b}-\frac {d \cosh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\)

Input:

Int[(c + d*x)^3*Cosh[a + b*x]^3,x]
 

Output:

-1/3*(d*(c + d*x)^2*Cosh[a + b*x]^3)/b^2 + ((c + d*x)^3*Cosh[a + b*x]^2*Si 
nh[a + b*x])/(3*b) + (2*d^2*(-1/9*(d*Cosh[a + b*x]^3)/b^2 + ((c + d*x)*Cos 
h[a + b*x]^2*Sinh[a + b*x])/(3*b) + (2*(-((d*Cosh[a + b*x])/b^2) + ((c + d 
*x)*Sinh[a + b*x])/b))/3))/(3*b^2) + (2*(((c + d*x)^3*Sinh[a + b*x])/b + ( 
(3*I)*d*((I*(c + d*x)^2*Cosh[a + b*x])/b - ((2*I)*d*(-((d*Cosh[a + b*x])/b 
^2) + ((c + d*x)*Sinh[a + b*x])/b))/b))/b))/3
 

Defintions of rubi rules used

rule 26
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3792
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 
Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.62

method result size
parallelrisch \(\frac {126 b^{2} d^{2} x \left (\frac {d x}{2}+c \right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-54 \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}+\frac {14 d^{2}}{3}\right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}+\left (\left (-27 d^{3} x^{2}-54 c \,d^{2} x +162 d \,c^{2}\right ) b^{2}+252 d^{3}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+36 \left (\left (d x +c \right )^{2} b^{2}+\frac {38 d^{2}}{3}\right ) \left (d x +c \right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\left (\left (-27 d^{3} x^{2}-54 c \,d^{2} x -216 d \,c^{2}\right ) b^{2}-480 d^{3}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-54 \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}+\frac {14 d^{2}}{3}\right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+\left (63 d^{3} x^{2}+126 c \,d^{2} x +126 d \,c^{2}\right ) b^{2}+244 d^{3}}{27 b^{4} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}\) \(284\)
risch \(\frac {\left (9 b^{3} d^{3} x^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x -9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}-18 b^{2} c \,d^{2} x -9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-2 d^{3}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}+\frac {3 \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{8 b^{4}}-\frac {3 \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}-\frac {\left (9 b^{3} d^{3} x^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x +9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}+18 b^{2} c \,d^{2} x +9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+2 d^{3}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\) \(415\)
orering \(-\frac {20 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}+22 b^{2} d^{4} x^{2}+44 b^{2} c \,d^{3} x +22 b^{2} c^{2} d^{2}-72 d^{4}\right ) \cosh \left (b x +a \right )^{3}}{27 b^{6} \left (d x +c \right )^{2}}+\frac {10 \left (3 d^{4} x^{4} b^{4}+12 b^{4} c \,d^{3} x^{3}+18 b^{4} c^{2} d^{2} x^{2}+12 b^{4} c^{3} d x +3 b^{4} c^{4}+2 b^{2} d^{4} x^{2}+4 b^{2} c \,d^{3} x +2 b^{2} c^{2} d^{2}-84 d^{4}\right ) \left (3 \left (d x +c \right )^{2} \cosh \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{3} \cosh \left (b x +a \right )^{2} b \sinh \left (b x +a \right )\right )}{27 \left (d x +c \right )^{4} b^{6}}+\frac {4 d \left (9 d^{2} x^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}+50 d^{2}\right ) \left (6 \left (d x +c \right ) \cosh \left (b x +a \right )^{3} d^{2}+18 \left (d x +c \right )^{2} \cosh \left (b x +a \right )^{2} d b \sinh \left (b x +a \right )+6 \left (d x +c \right )^{3} \cosh \left (b x +a \right ) b^{2} \sinh \left (b x +a \right )^{2}+3 \left (d x +c \right )^{3} \cosh \left (b x +a \right )^{3} b^{2}\right )}{27 b^{6} \left (d x +c \right )^{3}}-\frac {\left (3 d^{2} x^{2} b^{2}+6 b^{2} c d x +3 b^{2} c^{2}+20 d^{2}\right ) \left (6 d^{3} \cosh \left (b x +a \right )^{3}+54 \left (d x +c \right ) \cosh \left (b x +a \right )^{2} d^{2} b \sinh \left (b x +a \right )+54 \left (d x +c \right )^{2} \cosh \left (b x +a \right ) d \,b^{2} \sinh \left (b x +a \right )^{2}+27 \left (d x +c \right )^{2} \cosh \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{3} b^{3} \sinh \left (b x +a \right )^{3}+21 \left (d x +c \right )^{3} \cosh \left (b x +a \right )^{2} b^{3} \sinh \left (b x +a \right )\right )}{27 \left (d x +c \right )^{2} b^{6}}\) \(578\)
derivativedivides \(\frac {\frac {d^{3} \left (\frac {2 \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+\frac {40 \left (b x +a \right ) \sinh \left (b x +a \right )}{9}-\frac {40 \cosh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{9}-\frac {2 \cosh \left (b x +a \right )^{3}}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{27}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{27}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}+c^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}\) \(634\)
default \(\frac {\frac {d^{3} \left (\frac {2 \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+\frac {40 \left (b x +a \right ) \sinh \left (b x +a \right )}{9}-\frac {40 \cosh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{9}-\frac {2 \cosh \left (b x +a \right )^{3}}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{27}\right )}{b^{3}}+\frac {3 d^{2} c \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{9}+\frac {2 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{27}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{3}}-\frac {6 d^{2} a c \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\cosh \left (b x +a \right )^{3}}{9}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}+c^{3} \left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{b}\) \(634\)

Input:

int((d*x+c)^3*cosh(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/27*(126*b^2*d^2*x*(1/2*d*x+c)*tanh(1/2*b*x+1/2*a)^6-54*(d*x+c)*((d*x+c)^ 
2*b^2+14/3*d^2)*b*tanh(1/2*b*x+1/2*a)^5+((-27*d^3*x^2-54*c*d^2*x+162*c^2*d 
)*b^2+252*d^3)*tanh(1/2*b*x+1/2*a)^4+36*((d*x+c)^2*b^2+38/3*d^2)*(d*x+c)*b 
*tanh(1/2*b*x+1/2*a)^3+((-27*d^3*x^2-54*c*d^2*x-216*c^2*d)*b^2-480*d^3)*ta 
nh(1/2*b*x+1/2*a)^2-54*(d*x+c)*((d*x+c)^2*b^2+14/3*d^2)*b*tanh(1/2*b*x+1/2 
*a)+(63*d^3*x^2+126*c*d^2*x+126*c^2*d)*b^2+244*d^3)/b^4/(tanh(1/2*b*x+1/2* 
a)-1)^3/(tanh(1/2*b*x+1/2*a)+1)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (161) = 322\).

Time = 0.09 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.96 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=-\frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )^{3} + 243 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) - 9 \, {\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{3} + 54 \, b c d^{2} + {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 27 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \] Input:

integrate((d*x+c)^3*cosh(b*x+a)^3,x, algorithm="fricas")
 

Output:

-1/108*((9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*cosh(b*x + 
a)^3 + 3*(9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*cosh(b*x + 
 a)*sinh(b*x + a)^2 - 3*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 + 2*b 
*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*sinh(b*x + a)^3 + 243*(b^2*d^3*x^2 + 2 
*b^2*c*d^2*x + b^2*c^2*d + 2*d^3)*cosh(b*x + a) - 9*(9*b^3*d^3*x^3 + 27*b^ 
3*c*d^2*x^2 + 9*b^3*c^3 + 54*b*c*d^2 + (3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 
3*b^3*c^3 + 2*b*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*cosh(b*x + a)^2 + 27*(b 
^3*c^2*d + 2*b*d^3)*x)*sinh(b*x + a))/b^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).

Time = 0.51 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.83 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\begin {cases} - \frac {2 c^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 d^{3} x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac {2 c^{2} d \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {7 c^{2} d \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{2} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {14 c d^{2} x \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {7 d^{3} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac {40 c d^{2} \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 c d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {40 d^{3} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 d^{3} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {40 d^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} - \frac {122 d^{3} \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

integrate((d*x+c)**3*cosh(b*x+a)**3,x)
 

Output:

Piecewise((-2*c**3*sinh(a + b*x)**3/(3*b) + c**3*sinh(a + b*x)*cosh(a + b* 
x)**2/b - 2*c**2*d*x*sinh(a + b*x)**3/b + 3*c**2*d*x*sinh(a + b*x)*cosh(a 
+ b*x)**2/b - 2*c*d**2*x**2*sinh(a + b*x)**3/b + 3*c*d**2*x**2*sinh(a + b* 
x)*cosh(a + b*x)**2/b - 2*d**3*x**3*sinh(a + b*x)**3/(3*b) + d**3*x**3*sin 
h(a + b*x)*cosh(a + b*x)**2/b + 2*c**2*d*sinh(a + b*x)**2*cosh(a + b*x)/b* 
*2 - 7*c**2*d*cosh(a + b*x)**3/(3*b**2) + 4*c*d**2*x*sinh(a + b*x)**2*cosh 
(a + b*x)/b**2 - 14*c*d**2*x*cosh(a + b*x)**3/(3*b**2) + 2*d**3*x**2*sinh( 
a + b*x)**2*cosh(a + b*x)/b**2 - 7*d**3*x**2*cosh(a + b*x)**3/(3*b**2) - 4 
0*c*d**2*sinh(a + b*x)**3/(9*b**3) + 14*c*d**2*sinh(a + b*x)*cosh(a + b*x) 
**2/(3*b**3) - 40*d**3*x*sinh(a + b*x)**3/(9*b**3) + 14*d**3*x*sinh(a + b* 
x)*cosh(a + b*x)**2/(3*b**3) + 40*d**3*sinh(a + b*x)**2*cosh(a + b*x)/(9*b 
**4) - 122*d**3*cosh(a + b*x)**3/(27*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d 
*x**2/2 + c*d**2*x**3 + d**3*x**4/4)*cosh(a)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (161) = 322\).

Time = 0.08 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.51 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {1}{24} \, c^{2} d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} + \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{3} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{72} \, c d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} + \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} + \frac {1}{216} \, d^{3} {\left (\frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{4}} + \frac {81 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{4}}\right )} \] Input:

integrate((d*x+c)^3*cosh(b*x+a)^3,x, algorithm="maxima")
 

Output:

1/24*c^2*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 + 27*(b*x*e^a - e^a)*e 
^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x - a)/b^2 - (3*b*x + 1)*e^(-3*b*x - 3*a)/ 
b^2) + 1/24*c^3*(e^(3*b*x + 3*a)/b + 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b - 
e^(-3*b*x - 3*a)/b) + 1/72*c*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e 
^(3*a))*e^(3*b*x)/b^3 + 81*(b^2*x^2*e^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 
 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 - (9*b^2*x^2 + 6*b*x + 2)*e^(-3 
*b*x - 3*a)/b^3) + 1/216*d^3*((9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b 
*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x)/b^4 + 81*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 
6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 - 81*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(- 
b*x - a)/b^4 - (9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (161) = 322\).

Time = 0.14 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.37 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x - 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} - 18 \, b^{2} c d^{2} x - 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 2 \, d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} + \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x + 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 2 \, d^{3}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \] Input:

integrate((d*x+c)^3*cosh(b*x+a)^3,x, algorithm="giac")
 

Output:

1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 27*b^3*c^2*d*x - 9*b^2*d^3*x^2 + 
 9*b^3*c^3 - 18*b^2*c*d^2*x - 9*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 2*d^3) 
*e^(3*b*x + 3*a)/b^4 + 3/8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x 
- 3*b^2*d^3*x^2 + b^3*c^3 - 6*b^2*c*d^2*x - 3*b^2*c^2*d + 6*b*d^3*x + 6*b* 
c*d^2 - 6*d^3)*e^(b*x + a)/b^4 - 3/8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^ 
3*c^2*d*x + 3*b^2*d^3*x^2 + b^3*c^3 + 6*b^2*c*d^2*x + 3*b^2*c^2*d + 6*b*d^ 
3*x + 6*b*c*d^2 + 6*d^3)*e^(-b*x - a)/b^4 - 1/216*(9*b^3*d^3*x^3 + 27*b^3* 
c*d^2*x^2 + 27*b^3*c^2*d*x + 9*b^2*d^3*x^2 + 9*b^3*c^3 + 18*b^2*c*d^2*x + 
9*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 2*d^3)*e^(-3*b*x - 3*a)/b^4
 

Mupad [B] (verification not implemented)

Time = 2.18 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.08 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (3\,b^2\,c^3+14\,c\,d^2\right )}{3\,b^3}-\frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3+20\,c\,d^2\right )}{9\,b^3}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d+122\,d^3\right )}{27\,b^4}+\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^4}-\frac {2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^3}-\frac {7\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}-\frac {2\,d^3\,x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}-\frac {14\,c\,d^2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d+14\,d^3\right )}{3\,b^3}+\frac {d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {2\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2}-\frac {2\,c\,d^2\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{b}+\frac {3\,c\,d^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {4\,c\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2} \] Input:

int(cosh(a + b*x)^3*(c + d*x)^3,x)
 

Output:

(cosh(a + b*x)^2*sinh(a + b*x)*(14*c*d^2 + 3*b^2*c^3))/(3*b^3) - (2*sinh(a 
 + b*x)^3*(20*c*d^2 + 3*b^2*c^3))/(9*b^3) - (cosh(a + b*x)^3*(122*d^3 + 63 
*b^2*c^2*d))/(27*b^4) + (2*cosh(a + b*x)*sinh(a + b*x)^2*(20*d^3 + 9*b^2*c 
^2*d))/(9*b^4) - (2*x*sinh(a + b*x)^3*(20*d^3 + 9*b^2*c^2*d))/(9*b^3) - (7 
*d^3*x^2*cosh(a + b*x)^3)/(3*b^2) - (2*d^3*x^3*sinh(a + b*x)^3)/(3*b) - (1 
4*c*d^2*x*cosh(a + b*x)^3)/(3*b^2) + (x*cosh(a + b*x)^2*sinh(a + b*x)*(14* 
d^3 + 9*b^2*c^2*d))/(3*b^3) + (d^3*x^3*cosh(a + b*x)^2*sinh(a + b*x))/b + 
(2*d^3*x^2*cosh(a + b*x)*sinh(a + b*x)^2)/b^2 - (2*c*d^2*x^2*sinh(a + b*x) 
^3)/b + (3*c*d^2*x^2*cosh(a + b*x)^2*sinh(a + b*x))/b + (4*c*d^2*x*cosh(a 
+ b*x)*sinh(a + b*x)^2)/b^2
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 678, normalized size of antiderivative = 3.87 \[ \int (c+d x)^3 \cosh ^3(a+b x) \, dx=\frac {81 e^{4 b x +4 a} b^{3} d^{3} x^{3}-243 e^{4 b x +4 a} b^{2} c^{2} d -243 e^{4 b x +4 a} b^{2} d^{3} x^{2}+486 e^{4 b x +4 a} b c \,d^{2}+486 e^{4 b x +4 a} b \,d^{3} x -27 b^{3} c^{2} d x -27 b^{3} c \,d^{2} x^{2}-18 b^{2} c \,d^{2} x +9 e^{6 b x +6 a} b^{3} c^{3}-81 e^{2 b x +2 a} b^{3} c^{3}-2 d^{3}+9 e^{6 b x +6 a} b^{3} d^{3} x^{3}-9 e^{6 b x +6 a} b^{2} c^{2} d -9 e^{6 b x +6 a} b^{2} d^{3} x^{2}+6 e^{6 b x +6 a} b c \,d^{2}+6 e^{6 b x +6 a} b \,d^{3} x -81 e^{2 b x +2 a} b^{3} d^{3} x^{3}-243 e^{2 b x +2 a} b^{2} c^{2} d -243 e^{2 b x +2 a} b^{2} d^{3} x^{2}-486 e^{2 b x +2 a} b c \,d^{2}-486 e^{2 b x +2 a} b \,d^{3} x +81 e^{4 b x +4 a} b^{3} c^{3}-9 b^{3} d^{3} x^{3}-9 b^{2} c^{2} d -9 b^{2} d^{3} x^{2}-6 b c \,d^{2}-6 b \,d^{3} x -2 e^{6 b x +6 a} d^{3}+243 e^{4 b x +4 a} b^{3} c^{2} d x +243 e^{4 b x +4 a} b^{3} c \,d^{2} x^{2}-486 e^{4 b x +4 a} b^{2} c \,d^{2} x +27 e^{6 b x +6 a} b^{3} c^{2} d x +27 e^{6 b x +6 a} b^{3} c \,d^{2} x^{2}-18 e^{6 b x +6 a} b^{2} c \,d^{2} x -243 e^{2 b x +2 a} b^{3} c^{2} d x -243 e^{2 b x +2 a} b^{3} c \,d^{2} x^{2}-486 e^{2 b x +2 a} b^{2} c \,d^{2} x -486 e^{4 b x +4 a} d^{3}-9 b^{3} c^{3}-486 e^{2 b x +2 a} d^{3}}{216 e^{3 b x +3 a} b^{4}} \] Input:

int((d*x+c)^3*cosh(b*x+a)^3,x)
 

Output:

(9*e**(6*a + 6*b*x)*b**3*c**3 + 27*e**(6*a + 6*b*x)*b**3*c**2*d*x + 27*e** 
(6*a + 6*b*x)*b**3*c*d**2*x**2 + 9*e**(6*a + 6*b*x)*b**3*d**3*x**3 - 9*e** 
(6*a + 6*b*x)*b**2*c**2*d - 18*e**(6*a + 6*b*x)*b**2*c*d**2*x - 9*e**(6*a 
+ 6*b*x)*b**2*d**3*x**2 + 6*e**(6*a + 6*b*x)*b*c*d**2 + 6*e**(6*a + 6*b*x) 
*b*d**3*x - 2*e**(6*a + 6*b*x)*d**3 + 81*e**(4*a + 4*b*x)*b**3*c**3 + 243* 
e**(4*a + 4*b*x)*b**3*c**2*d*x + 243*e**(4*a + 4*b*x)*b**3*c*d**2*x**2 + 8 
1*e**(4*a + 4*b*x)*b**3*d**3*x**3 - 243*e**(4*a + 4*b*x)*b**2*c**2*d - 486 
*e**(4*a + 4*b*x)*b**2*c*d**2*x - 243*e**(4*a + 4*b*x)*b**2*d**3*x**2 + 48 
6*e**(4*a + 4*b*x)*b*c*d**2 + 486*e**(4*a + 4*b*x)*b*d**3*x - 486*e**(4*a 
+ 4*b*x)*d**3 - 81*e**(2*a + 2*b*x)*b**3*c**3 - 243*e**(2*a + 2*b*x)*b**3* 
c**2*d*x - 243*e**(2*a + 2*b*x)*b**3*c*d**2*x**2 - 81*e**(2*a + 2*b*x)*b** 
3*d**3*x**3 - 243*e**(2*a + 2*b*x)*b**2*c**2*d - 486*e**(2*a + 2*b*x)*b**2 
*c*d**2*x - 243*e**(2*a + 2*b*x)*b**2*d**3*x**2 - 486*e**(2*a + 2*b*x)*b*c 
*d**2 - 486*e**(2*a + 2*b*x)*b*d**3*x - 486*e**(2*a + 2*b*x)*d**3 - 9*b**3 
*c**3 - 27*b**3*c**2*d*x - 27*b**3*c*d**2*x**2 - 9*b**3*d**3*x**3 - 9*b**2 
*c**2*d - 18*b**2*c*d**2*x - 9*b**2*d**3*x**2 - 6*b*c*d**2 - 6*b*d**3*x - 
2*d**3)/(216*e**(3*a + 3*b*x)*b**4)